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Properties

Label 5687.2.a.bf
Level $5687$
Weight $2$
Character orbit 5687.a
Self dual yes
Analytic conductor $45.411$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5687,2,Mod(1,5687)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5687, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5687.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5687 = 11^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5687.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(45.4109236295\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 517)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 10 q^{2} + q^{3} + 46 q^{4} + 6 q^{5} + 15 q^{6} + 30 q^{7} + 27 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 10 q^{2} + q^{3} + 46 q^{4} + 6 q^{5} + 15 q^{6} + 30 q^{7} + 27 q^{8} + 37 q^{9} + 15 q^{10} - 9 q^{12} + 38 q^{13} + 5 q^{14} - 15 q^{15} + 42 q^{16} + 35 q^{17} + 29 q^{18} + 24 q^{19} + 32 q^{20} + 29 q^{21} + 5 q^{23} + 15 q^{24} + 26 q^{25} - 40 q^{26} + 7 q^{27} + 49 q^{28} + 18 q^{29} - 25 q^{30} + 19 q^{31} + 54 q^{32} - 11 q^{34} - q^{35} + 59 q^{36} - 3 q^{37} - 2 q^{38} + 16 q^{39} + 49 q^{40} + 28 q^{41} + 126 q^{42} + 78 q^{43} + 13 q^{45} + 57 q^{46} + 44 q^{47} - 29 q^{48} + 30 q^{49} + 31 q^{50} + 76 q^{51} + 67 q^{52} + 5 q^{53} + 44 q^{54} + 21 q^{56} + 66 q^{57} + q^{58} - q^{59} - 114 q^{60} + 48 q^{61} - 9 q^{62} + 91 q^{63} + 45 q^{64} + 58 q^{65} - 3 q^{67} + 55 q^{68} - 2 q^{69} - 50 q^{70} - 16 q^{71} + 54 q^{72} + 97 q^{73} + 2 q^{74} - 2 q^{75} + 65 q^{76} - 13 q^{78} + 52 q^{79} + 47 q^{80} - 36 q^{81} - 47 q^{82} + 28 q^{83} + 148 q^{84} + 69 q^{85} + 97 q^{86} - 43 q^{87} - 9 q^{89} + 81 q^{90} + 61 q^{91} - 36 q^{92} - 81 q^{93} + 10 q^{94} + 46 q^{95} + 5 q^{96} - 10 q^{97} + 83 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.59984 −2.49312 4.75919 −0.601486 6.48173 0.994149 −7.17347 3.21566 1.56377
1.2 −2.58571 0.124472 4.68591 −1.19898 −0.321848 1.10091 −6.94499 −2.98451 3.10023
1.3 −2.44906 −0.871087 3.99787 3.97951 2.13334 3.69062 −4.89290 −2.24121 −9.74605
1.4 −2.29159 1.83689 3.25136 −3.54309 −4.20938 −1.58326 −2.86761 0.374150 8.11930
1.5 −2.21170 −2.03665 2.89163 2.05766 4.50446 −0.830386 −1.97203 1.14793 −4.55094
1.6 −1.95767 −3.12657 1.83246 −1.92046 6.12080 4.61966 0.327979 6.77547 3.75962
1.7 −1.86479 2.49470 1.47745 0.00452873 −4.65209 3.27231 0.974451 3.22352 −0.00844514
1.8 −1.84301 2.77056 1.39667 −0.204252 −5.10616 4.01039 1.11194 4.67602 0.376437
1.9 −1.79511 −0.914836 1.22243 1.89259 1.64223 −3.37417 1.39583 −2.16307 −3.39740
1.10 −1.76381 1.17590 1.11101 −0.0242871 −2.07405 −3.62995 1.56801 −1.61727 0.0428377
1.11 −1.62871 −0.527914 0.652704 −0.382439 0.859820 0.299982 2.19436 −2.72131 0.622883
1.12 −1.06708 −2.33826 −0.861350 −3.32301 2.49509 −0.726574 3.05328 2.46744 3.54590
1.13 −0.964383 1.69116 −1.06996 1.64377 −1.63093 0.606677 2.96062 −0.139962 −1.58523
1.14 −0.832397 0.251243 −1.30712 4.12317 −0.209134 −0.959810 2.75283 −2.93688 −3.43211
1.15 −0.830951 −2.80840 −1.30952 2.95875 2.33365 3.97253 2.75005 4.88714 −2.45857
1.16 −0.806373 −1.10294 −1.34976 1.20409 0.889380 0.315257 2.70116 −1.78353 −0.970949
1.17 −0.542588 −1.40055 −1.70560 −4.04426 0.759923 3.46770 2.01061 −1.03845 2.19437
1.18 −0.457476 2.01968 −1.79072 −1.26334 −0.923956 −3.74787 1.73416 1.07912 0.577946
1.19 −0.295730 0.960285 −1.91254 −3.06724 −0.283985 0.813431 1.15706 −2.07785 0.907076
1.20 −0.0791297 0.162780 −1.99374 −2.86576 −0.0128808 0.362275 0.316023 −2.97350 0.226767
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(47\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5687.2.a.bf 44
11.b odd 2 1 5687.2.a.be 44
11.c even 5 2 517.2.e.e ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
517.2.e.e ✓ 88 11.c even 5 2
5687.2.a.be 44 11.b odd 2 1
5687.2.a.bf 44 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5687))\):

\( T_{2}^{44} - 10 T_{2}^{43} - 17 T_{2}^{42} + 481 T_{2}^{41} - 633 T_{2}^{40} - 10151 T_{2}^{39} + \cdots + 116 \) Copy content Toggle raw display
\( T_{3}^{44} - T_{3}^{43} - 84 T_{3}^{42} + 80 T_{3}^{41} + 3260 T_{3}^{40} - 2956 T_{3}^{39} + \cdots - 369724 \) Copy content Toggle raw display
\( T_{5}^{44} - 6 T_{5}^{43} - 105 T_{5}^{42} + 679 T_{5}^{41} + 4908 T_{5}^{40} - 34952 T_{5}^{39} + \cdots - 6004 \) Copy content Toggle raw display